sste39590_ch04S_001-019.inddte39590_ch04S_001-019.indd PagePage 4S-14S-1 03/12/1403/12/14 7:037:03 PMPM useruser //204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles SUPPLEMENT TO CHAPTER 4 Reliability SUPPLEMENT OUTLINE Solved Problems Learning Defining Reliability and Stating Discussion and Review Objectives Two Ways of Using It Questions Internet Exercises After completing this supple- Finding Probability of Functioning ment, you should be able to: When Activated Problems Finding Probability of Functioning Mini-Case: Engineer Tank LO1 Define reliability and state two ways of using it. for a Given Length of Time Mini-Case: Sonar System Key Terms LO2 Find probability of functioning when activated, and explain the purpose of redundancy in a system. LO3 Find probability of functioning for a given length of time, and define failure rate per hour, mean time to failure, and availability. 3rd Pass sste39590_ch04S_001-019.inddte39590_ch04S_001-019.indd PagePage 4S-24S-2 03/12/1403/12/14 7:037:03 PMPM useruser //204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles 4S-2 PART 3 System Design LO1 Defining Reliability and Stating Two Ways of Using It reliability The ability of a Reliability is a measure of the ability of a product or part to perform its intended function under product or part to perform its a prescribed set of conditions. In effect, reliability is a probability . intended function under a Suppose that an item has a reliability of .90. This means that it has a 90 percent probability of prescribed set of conditions. functioning as intended. The probability that it will fail, i.e., its failure rate, is 1 - .90 = .10, or 10 percent. Hence, it is expected that, on average, 1 out of every 10 such items will fail or, equiva- lently, that the item will fail, on average, once in every 10 trials. Similarly, a reliability of .985 implies 15 failures per 1,000 parts or trials. Reliability of a product or part is used in two ways. 1. Reliability when activated. 2. Reliability for a given length of time. The first of these focuses on one point in time and is often used when a product or part must operate for one time, such as a missile or an air bag in a car. The second of these focuses on the length of service , such as most other products e.g., a car. The distinction will become more apparent as each of these approaches is described in more detail. Reliability is an important dimension of product quality. Reliability management involves establishing, achieving, and maintaining reliability objectives for products, e.g., the expected life of a particular make of light bulb may be specified to be 5,000 hours. Achieving reliability usually falls on the shoulder of reliability engineers who use a variety of techniques to build reliability into products (e.g., by using reliable key components), test their performance, and estimate their reliability. If the reliability is inadequate, the types of failure and their effect on the product should be determined, their root cause(s) identified, and potential failure prevented. We will mainly focus on reliability measurement, which involves statistics and probability theory. The average reliability of a part is measured by testing several units over time until some or all fail. However, this time may be very long (several years). To accelerate this, the items are stressed by using extreme environmental conditions such as high temperature, temperature cycles (e.g., hot–cold), high humidity, high vibration, high voltage, surges in power, etc. The result- ing life estimate is then adjusted appropriately. Reliability of a product is determined from the reliability of its parts. LO2 Finding Probability of Functioning When Activated The probability that a part or product will operate as planned is an important concept in product design. Determining that probability when the product consists of a number of independent com- ponents requires the use of rules of probability for independent events. Independent events have no relation to the occurrence or nonoccurrence of each other. What follows are examples illustrat- ing the use of two probability rules to determine whether a given product will operate successfully. Let Pi = probability that event i occurs, i = 1, 2, 3, . Rule 1. If two or more events are independent and “success” is defined as the occurrence of all of the events, then the probability of success P s is equal to the product of the probabilities of the events occurring, i.e., Ps = P1 × P 2 × . Example. Suppose a room has two lamps, but to have adequate light, both lamps must work (success) when turned on. Here the product is the lighting system that has two component lamps. One lamp has a probability of working of .90, and the other has a probability of working of .80. The probability that both will work is .90 × .80 = .72. This lighting system can be represented by the following diagram where the two components are connected in series: Lamp 1 Lamp 2 .90 .80 3rd Pass sste39590_ch04S_001-019.inddte39590_ch04S_001-019.indd PagePage 4S-34S-3 03/12/1403/12/14 7:037:03 PMPM useruser //204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles SUPPLEMENT TO CHAPTER 4 Reliability 4S-3 CR 0.9 CR 0.98 CR 0.99 Figure 4S-1 1.0 Relating product and component .9 reliabilities. .8 .7 .6 .5 .4 .3 Product reliability .2 .1 0 12 345 67 89101112131415161718192021222324252627282930 Number of components in product Even though the individual components of a series system (product) might have high reliabil- ity, the series system (product) as a whole can have considerably less reliability because all its components must function (i.e., the system is dependent on each of its components). As the num- ber of components in a series system (product) increases, the system (product) reliability decreases. For example, a series system (product) that has eight components, each with a reliability of .99, has a reliability of only .99 8 = .923. See Figure 4S-1 for plots of product reliability as a function of num- ber of its components for selected component reliability, CR . Many products have a large number of component parts that must all operate, and some way to increase overall reliability is needed. One approach is to overdesign, i.e., enhance the design to avoid a particular type of failure. For example, use a more durable and higher quality (but more expensive) material in a product. Another is design simplification, i.e., reduce the number of com- ponents in the product. The third approach is to use redundancy in the design. This involves redundancy The use of providing backup components. backup components to increase reliability. Rule 2. If two or more events are independent and “success” is defined as occurrence of at least one of the events, then the probability of success Ps is equal to 1 - probability that none of the events will occur, i.e., 1 - (1 - P1 )(1 - P2 )(1 - P3 ). Simplifying, P s = P1 + (1 - P1 ) P2 + (1 - P1 ) (1 - P2 ) P3 + · · · Example. There are two lamps in a room. When turned on, one has probability of working of .90 and the other has probability of working of .80. Only a single lamp is needed to light the room for success (note that the threshold for success is different in this example). Then, probability of suc- cess Ps = 1 - (1 - .90)(1 - .80) = .98. Conceptually, we can think of this system as a lamp with a backup . If the first lamp fails to light when turned on, the backup lamp is turned on. The probability of success Ps is probability that the first lamp operates plus probability that the first lamp fails and the backup lamp operates, i.e., .90 + (1 - .90) × .80 = .98. This backup system can be represented by the following diagram. .80 Lamp 2 (backup) .90 Lamp 1 Example. Three lamps have probabilities of .90, .80, and .70 of lighting when turned on. Only one lighted lamp is needed for success. Then, probability of success Ps = 1 - (1 - .90)(1 - .80)(1 - .70) = .994. 3rd Pass sste39590_ch04S_001-019.inddte39590_ch04S_001-019.indd PagePage 4S-44S-4 03/12/1403/12/14 7:037:03 PMPM useruser //204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles204/MHR00256/ste39590_disk1of1/0071339590/ste39590_pagefiles 4S-4 PART 3 System Design Conceptually, we can think of this system as a lamp with a backup which in turn has a backup. If the first lamp fails to light when turned on, the second lamp is turned on. If the sec- ond lamp also fails to light when turned on, the third lamp is turned on. The probability of suc- cess Ps is probability that the first lamp operates plus probability that the first lamp fails and the second lamp operates plus probability that the first and second lamps fail and the third lamp operates, i.e.: 3#1 operates4 + 3#1 fails and #2 operates4 + 3#1 fails and #2 fails and #3 operates4 .90 + 11 − .902 × .80 + 311 − .902 × 11 − .802 × .704 = .994 This double backup system can be represented by the following diagram: .70 Lamp 3 (backup for Lamp 2) .80 Lamp 2 (backup for Lamp 1) .90 Lamp 1 In general, a product (system) may be composed of some parallel components and some series components. The product’s reliability is calculated in two stages: (1) first calculate the reliability of the parallel component(s) and then (2) use these to calculate the reliability of the resulting series system. Example 1 Determine the reliability of the system shown below. .90 .92 .98 .90 .95 SOLUTION (a) The system can be reduced to a series of three components: .98 .90 ϩ .90(1 Ϫ .90) .95 ϩ (1 – .95).92 (b) The system reliability is, then, the product of these component reliabilities: .98 × .99 × .996 = .966 LO3 Finding Probability of Functioning for a Given Length of Time The second way of looking at reliability considers a use factor, usually the time dimension: prob- abilities are determined relative to a specified length of time.